Nicolas Hadjisavvas

Vector equilibrium problems with generalized monotone bifunctions

AbstractA vector equilibrium problem is defined as follows: given a closed convex subset K of a real topological Hausdorff vector space and a bifunction F(x, y) valued in a real ordered locally convex vector space, find x*∈K such that

From scalar to vector equilibrium problems in the quasimonotone case

F(x^* ,y) \nless 0

On Quasimonotone Variational Inequalities

for all y∈K. This problem generalizes the (scalar) equilibrium problem and the vector variational inequality problem. Extending very recent results for these two special cases, the paper establishes existence of solutions for the unifying model, assuming that F is either a pseudomonotone or quasimonotone bifunction.

Coercivity conditions and variational inequalities

The chapters are as follows: Introduction to Convex and Quasiconvex Analysis (J.B.G.Frenk, G. Kassay) Criteria for Generalized Convexity and Generalized Monotonicity in the Differentiable Case (J.-P. Crouzeix) Continuity and Differentiability of Quasiconvex Functions (J.-P. Crouzeix) Generalized Convexity and Optimality Conditions in Scalar and Vector Optimization (A. Cambini, L. Martein) Generalized Convexity in Vector Optimization (D. T. Luc) Generalized Convex Duality and Its Economic Applications (J.-E. Martinez-Legaz) Abstract Convexity (A.Rubinov, J.Dutta) Fractional programming (J.B.G. Frenk, S.Schaible) Generalized Monotone Maps (N.Hadjisavvas, S.Schaible) Generalized Convexity and Generalized Derivatives (S.Komlosi) Generalized Convexity, Generalized Monotonicity and Nonsmooth Analysis (N.Hadjisavvas) Pseudomonotone Complementarity Problems and Variational Inequalities (J.-C. Yao, O. Chadly) Generalized Monotone Equilibrium Problems and Variational Inequalities (I. Konnov) Uses of Generalized Convexity and Monotonicity in Economics (R. John)

Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems

In a unified approach, existence results for quasimonotone vector equilibrium problems and quasimonotone (multivalued) vector variational inequality problems are derived from an existence result for a scalar equilibrium problem involving two (rather than one) quasimonotone bifunctions. The results in the vector case are not only obtained in a new way, but they are also stronger versions of earlier existence results.

EXISTENCE THEOREMS FOR VECTOR VARIATIONAL INEQUALITIES

Various existence results for variational inequalities in Banach spaces are derived, extending some recent results by Cottle and Yao. Generalized monotonicity as well as continuity assumptions on the operatorf are weakened and, in some results, the regularity assumptions on the domain off are relaxed significantly. The concept of inner point for subsets of Banach spaces proves to be useful.

Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions

The purpose of this paper is to prove the existence of solutions of the Stampacchia variational inequality for a quasimonotone multivalued operator without any assumption on the existence of inner points. Moreover, the operator is not supposed to be bounded valued. The result strengthens a variety of other results in the literature.

Quasimonotonicity and Pseudomonotonicity in Variational Inequalities and Equilibrium Problems

Various coercivity conditions appear in the literature in order to guarantee solutions for the Variational Inequality Problem. We show that these conditions are equivalent to each other and that they are not only sufficient, but also necessary for the set of solutions to be non-empty and bounded.